Cartan decomposition

The Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition of matrices.

1 Cartan involutions on Lie algebras
- 1.1 Examples
2 Cartan pairs
3 Cartan decomposition on the Lie group level
4 Relation to polar decomposition
5 See also
6 References

Cartan involutions on Lie algebras

Let $\mathfrak{g}$ be a real semisimple Lie algebra and let $B(\cdot,\cdot)$ be its Killing form. An involution on $\mathfrak{g}$ is a Lie algebra automorphism $\theta$ of $\mathfrak{g}$ whose square is equal to the identity. Such an involution is called a Cartan involution on $\mathfrak{g}$ if $B_\theta(X,Y) = -B(X,\theta Y)$ is a positive definite bilinear form.

Two involutions $\theta_1$ and $\theta_2$ are considered equivalent if they differ only by an inner automorphism.

Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.

Examples

A Cartan involution on $\mathfrak{sl}_n(\mathbb{R})$ is defined by $\theta(X)=-X^T$ , where $X^T$ denotes the transpose matrix of $X$ .

The identity map on $\mathfrak{g}$ is an involution, of course. It is the unique Cartan involution of $\mathfrak{g}$ if and only if the Killing form of $\mathfrak{g}$ is negative definite. Equivalently, $\mathfrak{g}$ is the Lie algebra of a compact Lie group.

Let $\mathfrak{g}$ be the complexification of a real semisimple Lie algebra $\mathfrak{g}_0$ , then complex conjugation on $\mathfrak{g}$ is an involution on $\mathfrak{g}$ . This is the Cartan involution on $\mathfrak{g}$ if and only if $\mathfrak{g}_0$ is the Lie algebra of a compact Lie group.

The following maps are involutions of the Lie algebra $\mathfrak{su}(n)$ of the special unitary group SU(n):

- the identity involution $\theta_0(X) = X$ , which is the unique Cartan involution in this case;

- $\theta_1 (X) = - X^T$ which on $\mathfrak{su}(n)$ is also the complex conjugation;

- if $n = p%2Bq$ is odd, $\theta_2 (X) = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix} X \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}$ . These are all equivalent, but not equivalent to the identity involution (because the matrix $\begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}$ does not belong to $\mathfrak{su}(n)$ .)

- if $n = 2m$ is even, we also have $\theta_3 (X) = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix} X^T \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix}.$

Cartan pairs

Let $\theta$ be an involution on a Lie algebra $\mathfrak{g}$ . Since $\theta^2=1$ , the linear map $\theta$ has the two eigenvalues $\pm1$ . Let $\mathfrak{k}$ and $\mathfrak{p}$ be the corresponding eigenspaces, then $\mathfrak{g} = \mathfrak{k}%2B\mathfrak{p}$ . Since $\theta$ is a Lie algebra automorphism, eigenvalues are multiplicative. It follows that

$[\mathfrak{k}, \mathfrak{k}] \subseteq \mathfrak{k}$ , $[\mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p}$ , and $[\mathfrak{p}, \mathfrak{p}] \subseteq \mathfrak{k}$ .

Thus $\mathfrak{k}$ is a Lie subalgebra, while any subalgebra of $\mathfrak{p}$ is commutative.

Conversely, a decomposition $\mathfrak{g} = \mathfrak{k}%2B\mathfrak{p}$ with these extra properties determines an involution $\theta$ on $\mathfrak{g}$ that is $%2B1$ on $\mathfrak{k}$ and $-1$ on $\mathfrak{p}$ .

Such a pair $(\mathfrak{k}, \mathfrak{p})$ is also called a Cartan pair of $\mathfrak{g}$ .

The decomposition $\mathfrak{g} = \mathfrak{k}%2B\mathfrak{p}$ associated to a Cartan involution is called a Cartan decomposition of $\mathfrak{g}$ . The special feature of a Cartan decomposition is that the Killing form is negative definite on $\mathfrak{k}$ and positive definite on $\mathfrak{p}$ . Furthermore, $\mathfrak{k}$ and $\mathfrak{p}$ are orthogonal complements of each other with respect to the Killing form on $\mathfrak{g}$ .

Cartan decomposition on the Lie group level

Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Let $\theta$ be a Cartan involution on $\mathfrak{g}$ and let $(\mathfrak{k},\mathfrak{p})$ be the resulting Cartan pair. Let $K$ be the analytic subgroup of $G$ with Lie algebra $\mathfrak{k}$ . Then

There is a Lie group automorphism $\Theta$ with differential $\theta$ that satisfies $\Theta^2=1$ .
The subgroup of elements fixed by $\Theta$ is $K$ ; in particular, $K$ is a closed subgroup.
The mapping $K\times\mathfrak{p} \rightarrow G$ given by $(k,X) \mapsto k\cdot \mathrm{exp}(X)$ is a diffeomorphism.
The subgroup $K$ contains the center $Z$ of $G$ , and $K$ is compact modulo center, that is, $K/Z$ is compact.
The subgroup $K$ is the maximal subgroup of $G$ that contains the center and is compact modulo center.

The automorphism $\Theta$ is also called global Cartan involution, and the diffeomorphism $K\times\mathfrak{p} \rightarrow G$ is called global Cartan decomposition.

For the general linear group, we get $X \mapsto (X^{-1})^T$ as the Cartan involution.

Relation to polar decomposition

Consider $\mathfrak{gl}_n(\mathbb{R})$ with the Cartan involution $\theta(X)=-X^T$ . Then $\mathfrak{k}=\mathfrak{so}_n(\mathbb{R})$ is the real Lie algebra of skew-symmetric matrices, so that $K=\mathrm{SO}(n)$ , while $\mathfrak{p}$ is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from $\mathfrak{p}$ onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix. Notice that the polar decomposition of an invertible matrix is unique.

References

A. W. Knapp, Lie groups beyond an introduction, ISBN 0-8176-4259-5, Birkhäuser.